Extraction of feature lines on triangulated surfaces using morphological operators

Triangle meshes are a popular representation of surfaces in computer graphics. Our aim is to detect feature on such surfaces. Feature regions distinguish themselves by high curvature. We are using discrete curvature analysis on triangle meshes to obtain curvature values in every vertex of a mesh. These values are then thresholded resulting in a so called binary feature vector. By adapting morphological operators to triangle meshes, noise and artifacts can be removed from the feature. We introduce an operator that determines the skeleton of the feature region. This skeleton can then be converted into a graph representing the desired feature. Therefore a description of the surface’s geometrical characteristics is constructed. Introduction There are several different representations for surfaces used in computer graphics, ranging e.g. from parametric surfaces defined as functions over a parameter domain to surfaces described by constructive solid geometry that allows combination of simple geometric primitives. As usual in computer science, different representations are favored for different purposes. Even so, there is a very simple but effective representation for many kinds of surfaces: triangle meshes. On the one hand they are capable of describing surfaces of arbitrary shape and topology. On the other hand they use the most basic graphic primitive for describing a surface, hence graphics hardware usually deals with sets of triangles. In recent years triangle meshes have grown to become popular, as for example techniques have been developed to use such meshes directly for geometric modeling in a comfortable way rather than the usual parametric free-form surfaces, e.g. NURBS (Kobbelt et. al. 1998). Besides, a triangulated surface may be obtained from a laser range scanner sampling a real world object. The huge sets of data which arise as a result can now be reduced without losing too much accuracy of the surface representation (Kobbelt, Campagna, and Seidel 1998). For this reason, triangle meshes are the most appropriate representation when dealing with real world objects. Copyright c 2000, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. This paper focuses on extracting a structured description of an object represented by a triangulated surface from its geometry. Thus, we enrich an unstructured geometric representation with semantics of the object being investigated. The result may then be used as input to other algorithms. Therefore we propose a technique to find feature lines on a triangle mesh. In feature regions the local geometry of the mesh is heavily bent. Feature lines approximately pass along the ridge of maximum inflection. Mathematically such local characteristics of geometry can be measured by the curvature of the surface. This kind of information is important for a variety of applications as e.g rating surface quality or classification of surfaces, patch layouting for reverse engineering, or avoidance of aliasing during remeshing by resampling the surface. Applying some binary threshold operation on the obtained curvature values leaves us with a feature vector whose components represent the binary values assigned to the vertices. We then adapt morphological operators from digital image analysis in order to reduce noise and irritating artifacts on the feature vector, and we construct operators to extract a “feature skeleton”. This skeleton can then be transformed into a graph afterwards. Its edges represent polygons on the mesh lying near the region of interest, i.e. the feature to be extracted. The polygons are connected by the nodes of this graph. Thus, we transform an unstructured surface representation into a more structured description of surface characteristics defined by geometry. This description may be used as input to further algorithms. Discrete curvature analysis We use geometric curvature to extract features of a surface. As curvature is invariant to rotations or translations of an object, one can expect similar results from (not too) different views of the object or at least some overlap of features. E.g. registration of different views or scans can take advantage of this fact. Curvature also allows to classify different regions of an object. So geometric primitives that have been used to construct the object may be recovered. Curvature cannot be evaluated directly for triangle meshes, because it is mathematically defined for smooth surfaces only. A triangle mesh is a piecewise linear surface, so it is not clear how to calculate any derivatives on such a mesh. Our approach locally estimates the first and second fundamental form of the surface F (u; v) in every vertex of its triangulation. Deriving surface curvatures like principle curvatures from the fundamental forms is straightforward. An introduction to the basic concepts of differential geometry can be found e.g. in (Farin 1996; do Carmo 1976). First, we construct a nearly isometric (kF